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Letters Super Water Repellent Fractal Surfaces T Onda S Shibuichi N Satoh and K Tsujii Kao Institute for Fundamental Research Kao Corporation 2606 Akabane Ichikai machi Haga gun Tochigi 321 34 Japan and Tokyo Research Laboratories Kao Corporation 2 1 3 Bunka Sumida ku Tokyo 131 Japan Received May 30 1995 In Final Form November 14 1995X Wettability of fractal surfaces has been studied both theoretically and experimentally The contact angle of a liquid droplet placed on a fractal surface is expressed as a function of the fractal dimension therangeoffractalbehavior andthecontactingratioofthesurface Theresultshowsthatfractalsurfaces canbesuperwaterrepellent superwettable whenthesurfacesarecomposedofhydrophobic hydrophilic materials We also demonstrate a super water repellent fractal surface made of alkylketene dimer a water droplet on this surface has a contact angle as large as 174 Ultrahydrophobic solid surfaces which perfectly repel water would bring great convenience on our daily life Conventionally thewettabilityofsolidsurfaceshasbeen controlledbychemicalmodificationsofthesurfaces such as fluorination 1The present Letter highlights the other factor determining the wettability i e the geometrical structureofsolidsurfaces 2 4Thereexists asMandelbrot hasemphasizedinhistextbook 5afascinatinggeometrical structure called fractal which is characterized by self similarity and a noninteger dimension Then a question arsies What is the wettability of the solid surface with a fractal structure This Letter shows both theoretically and experimentally that fractal surfaces can be either superrepellentorsuperwettabletoaliquid Furthermore we demonstrate a super water repellent fractal surface madeofalkylketenedimer awaterdropletonthissurface has a contact angle as large as 174 The contact angle of a liquid placed on a flat solid surface is given by Young s equation2 where R12 R13 and R23denote the interfacial tensions of the solid liquid the solid gas and the liquid gas interface respectively When a solid surface is rough Young s equation is modified into2 where r is a coefficient giving the ratio of the actual area of a rough surface to the projected area A fractal surface is a kind of rough surface so that the wettability of the fractal surface is basically described by eq 2 3The coefficient r of the fractal surface however is very large and can even be infinite for a mathematically ideal fractal surface Therefore the modification of the wettability due to surface roughness can be greatly enhancedinthefractalsurface 3thatis thefractalsurface will be superrepellent superwettable to a liquid when is greater less than 90 Strictly speaking applicability of eq 2 is limited In fact eq2cannotgiveanycontactanglewhentheabsolute value of its right hand side exceeds unity A correct expression for the contact angle can be derived conve niently by introducing the effective interfacial tension of Kao Institute for Fundamental Research Kao Corporation Tokyo Research Laboratories Kao Corporation XAbstractpublishedinAdvanceACSAbstracts March15 1996 1 Watanabe N Tei Y Kagaku 1991 46 477 in Japanese 2 Adamson A W Physical Chemistry of Surfaces 5th ed John Wiley Chapter X Section 4 3 Hazlett R D J Colloid Interface Sci 1990 137 527 4 Good R J Mikhail R S Powder Technol 1981 29 53 5 Mandelbrot B B The Fractal Geometry of Nature Freeman San Francisco CA 1982 cos R13 R12 R23 1 cos r r R13 R12 R23 r cos 2 Copyright 1996 American Chemical Society MAY 1 1996 VOLUME 12 NUMBER 9 S0743 7463 95 00418 5 CCC 12 00 1996 American Chemical Society the solid liquid fractal interface Rf12 and that of the solid gas fractal interface Rf13 Then the contact angle fis given by SinceRf1j j 2 3 maybeconsideredthetotalinterfacial energy of the fractal surface per unit projected area Rf1j can be estimated as a first approximation by Rf1j L l D 2R1j Here D 2 e D 3 is the fractal dimension of the surface L and l are respectively the upper and the lower limit lengths of fractal behavior Substitution of thisRf1jintoeq3yieldseq2inwhich L l D 2issubstituted for r To be more precise in estimating Rf1j we must take account of the adsorption of a gas on the solid liquid fractal interface and that of a liquid on the solid gas fractal interface Then Rf1jcan be expressed by where x 0e x e 1 is the fraction of the area of the fractal surfacecoveredwiththeadsorbedgas y 0eye1 isthat covered with the adsorbed liquid and S23is the area of the gas liquid interface per unit projected area The symbol min expresses the operation of minimizing the succeedingquantitieswithrespecttoxory thisoperation corresponds to the fact that adsorption so occurs as to minimize the total interfacial energy Typically S23 z z x y can be approximated by a cubic function of z given by Figure1 Schematicillustrationforcos fvscos theoretically predicted Figure2 SEMimagesofthefractalAKDsurface a top top view b bottom cross section cos f Rf13 Rf12 R23 3 Figure 3 Water droplet on AKD surfaces a top fractal AKD surface f 174 b bottom flat AKD surface 109 The diameter of the droplets is about 2 mm Rf12 min x R13 L l D 2 x R12 L l D 2 1 x R23S23 x 4 Rf13 min y R13 L l D 2 1 y R12 L l D 2 y R23S23 y 5 S23 z z L l D 2 1 z 1 z 1 1 3 1 2 z 6 2126Langmuir Vol 12 No 9 1996Letters where denotesthecontactingratioofthefractalsurface the ratio of the area touching a flat plate placed on the fractal surface Behavior of fdetermined by eqs 3 6 is shownschematicallyinFigure1 inwhichcos fisplotted as a function of cos R13 R12 R23 The obtained curve coincides with the line of cos f L l D 2cos in the vicinity of 90 but deviates from it owing to the adsorption when approaches 0 or 180 To verify the theoretical prediction we have experi mentally examined the wettability of various kinds of fractal surfaces In this Letter we report the results of the fractal surface made of alkylketene dimer AKD which is a kind of wax and one of the sizing agents for papers Alkylketenedimermostlyusedinourexperimentswas synthesized from stearoyl chloride with the use of tri ethylamine as a catalyst and purified up to 98 with a silica gel column The purity was checked by gel per meation chromatography and capillary gas chromatog raphy the major impurity was dialkyl ketone which was producedbythehydrolysisofAKD AKDsolidfilmswith a thickness of about 100 m were grown on glass plates a glass plate was dipped into melted AKD heated at 90 C and was then cooled at room temperature in the ambience of dry N2gas AKD undergoes fractal growth when it solidifies although the mechanism has not been clarified yet SEM images of the AKD solid surface are shown in Figure 2 To find the fractal dimension of the AKD surface we applied the box counting method to the cross section of one of the AKD films The AKD film peeled off from the glass plate was cleaved with the aid of a razor blade and SEM images of its cross section Figure 2b were taken at several magnifications Then the fractal dimension of the cross section Dcross has been measured by the box countingmethodandfoundtobe1 29intherangebetween l 0 2 mandL 34 m seetheinsetofFigure4 below andabovetherange Dcrossisfoundtobeunity Thus the fractal dimension of the AKD surface has been evaluated as D Dcross 1 2 29 Figure 3a shows a water droplet placed on the fractal AKD surface This surface as expected repels water completely and a contact angle as large as 174 has been obtained For comparison we have also examined the wettabilityofamechanicallyflattenedAKDsurface which was prepared by cutting a fractal AKD surface with a razor blade The flat AKD surface however does not repelwaterverymuch showingacontactanglenotlarger than 109 Figure 3b Comparison of parts a and b of Figure3highlightstheimportanceofthefractaleffecton the wettability The relationship between the contact angle for the fractalAKDsurface f andthatfortheflatAKDsurface has also been studied with the use of liquids of various surfacetensions Astheliquids weusedaqueoussolutions of1 4 dioxaneatvariousconcentrations becomessmaller as the concentration of 1 4 dioxane increases The result isshowninFigure4 inwhichcos fisplottedasafunction of cos The measurement of the contact angle has been performed several times at each concentration and the dispersion of the data is expressed by error bars The fractal AKD surface showed advancing and receding co